The Nyquist-Shannon sampling theorem may be stated as follows: For a bandlimited signal with maximum frequency of B Hz, the signal may be perfectly reconstructed from uniform samples of the signal if the samples are taken at a sampling rate that is greater than the Nyquist rate (2B samples per second).
Nyquist-Shannon sampling is often used in signal processing. For example, in many conventional sensing systems: (a) an analog sensor takes measurements of a physical phenomenon and outputs an analog electrical signal that encodes the measurements; and (b) an analog-to-digital converter (ADC) takes pointwise samples of the analog electrical signal at a sampling rate that is greater than the Nyquist rate.
Unfortunately, sampling at the Nyquist rate does not ensure accurate reconstruction of a signal if the dynamic range of the signal is too large. This is because, if the dynamic range is too large, an ordinary ADC may saturate or clip.
For example: (a) an ordinary ADC that is sampling an analog electrical signal may saturate once the input voltage reaches a saturation voltage; and (b) further increases in the input voltage (above the saturation level) may fail to produce a further increase in the sample value outputted by the ADC. Instead, for all input voltages that are at or above saturation voltage, the ADC may output substantially the same value (encoded in a digital signal).
When an ordinary ADC saturates, the resulting samples are clipped and hence the reconstruction is distorted and erroneous. The clipping is a serious problem that may manifest as non-linear artifacts in a variety of applications including audio-visual data, physiological data, or biomedical data). Clipping of bandlimited signals may result in discontinuities that comprise high-frequency distortions to the signal.
An attempted solution to the clipping problem has been to employ ADC's that effectively perform modulo operations, in such a manner that the values outputted by the ADC “wrap around” or “fold” when the input voltage reaches a certain value. Put loosely, these ADC's reset themselves when the input voltage reaches a certain value. Depending on the author and the field of endeavor, an ADC which performs modulo operations is sometimes called a folding ADC, reset ADC, self-reset ADC, SR-ADC, modulo ADC, or modulo PCM (pulse code modulator). For the remainder of this document, we shall refer to all of these as self-reset ADCs or SR-ADCs.
There has been relatively little research into how to recover an input signal from modulo samples acquired by a self-reset ADC. Conventional recovery methods that have been employed—to attempt to do this—suffer from at least two problems:
First, no one has known how to recover a bandlimited, continuous input signal from the modulo samples, in a manner that is mathematically guaranteed to recover the input signal (or that is mathematically guaranteed to recover an estimated signal that is equal to the input signal plus a constant bias signal).
Second, some conventional recovery methods for SR-ADCs have required knowledge of the number of folds that were employed to create each folded modulo sample, respectively. Thus, some conventional SR-ADCs are configured to count the number of folds. Recording the number of folds is often undesirable, because it may pose additional hardware constraints and require extra memory to store the number of folds.